Question Video: Determining the Properties of a Parallelogram Mathematics

Which of the following statements must be true about a parallelogram? [A] It has four sides of equal length. [B] It has four congruent sides. [C] It has four right angles. [D] It has one pair of parallel sides. [E] It has two pairs of parallel opposite sides.

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Video Transcript

Which of the following statements must be true about a parallelogram? Option (A) it has four sides of equal length. Option (B) it has four congruent sides. Option (C) it has four right angles. Option (D) it has one pair of parallel sides. Or option (E) it has two pairs of parallel opposite sides.

Here, we are considering different statements that may or may not be true about a parallelogram. So let’s recall that a parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel. This is what is given in the final answer option (E). If having two pairs of opposite sides parallel is what defines a parallelogram, then it must be true of every parallelogram. But let’s consider what might be happening in the other options. And to do this, let’s draw some different parallelograms.

Here, we have a very typical drawing of a parallelogram. It has got both pairs of opposite sides parallel, so it is indeed a parallelogram. And here, we have a different parallelogram with both pairs of opposite sides parallel. Now, it just so happens that this parallelogram also has four congruent sides. In fact, we could call this special type of parallelogram a rhombus, since this is defined as a parallelogram with four congruent sides or indeed with four sides of equal length.

The two statements in option (A) and (B) are equivalent and show the exact property that we have here. So why aren’t either of these the answer to the question? Well, it’s because although they do fit with a rhombus, which is a type of parallelogram, we can’t say that every parallelogram has four congruent sides or four sides of equal measure. And remember, we are looking for a statement which must be true about a parallelogram and indeed every parallelogram. So we can eliminate answer options (A) and (B).

Now, let’s consider option (C). As you might have guessed, we can draw a parallelogram that also has four right angles. And of course, this is a rectangle, which is defined as a parallelogram with four congruent angles. In the same reasoning as before, we can note that although this parallelogram, which is a rectangle, does have four right angles, not every parallelogram does. And since we are looking for a statement that is true about every parallelogram, then option (C) is incorrect.

Now, let’s consider option (D), which is the property that a parallelogram has one pair of parallel sides. It might be reasonable to say that if a quadrilateral has two pairs of parallel sides, then it has one pair of parallel sides. And if the answer option (E) wasn’t available, then it might be a good choice. However, it does sound a little more like the definition of a trapezoid, sometimes called a trapezium, depending on where you live. A trapezoid is defined as a quadrilateral with one pair, or exactly one pair, of parallel sides. Because of the fact that this statement may be confused with the property of having just one pair of parallel sides and the fact that we have this much better and more useful statement below that a parallelogram must have two pairs of parallel sides, then we can eliminate option (D).

Therefore, the statement that must be true about a parallelogram is that given in option (E). It has two pairs of parallel opposite sides.

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